Unfortunately, (weakly) smooth conditions typically can not cover
a mixture of distributions with different tail growth behaviors, which
prohibit a large range of applications. Therefore, we first introduce
an \(\alpha \) -mixture weakly smooth condition to overcome the limitation
of the current weakly smooth condition. Under our novel condition
and log-Sobolev inequality, we will recover the ULA convergence results
[1]}. In addition, we also show that ULA for
smoothing potential will converge in \(L_{2}\) -Wasserstein distance.
Since log-Sobolev inequality is preserved under bounded perturbation,
we will extend the results based on convexification of a non-convex
domain [2]}. Our contributions can be outlined
as follows.
